1. Chapter 4 Wave-Wave Interactions in Irregular Waves Irregular waves are approximately simulated by many periodic wave trains (free or linear waves) of different frequencies, amplitudes, and advancing in different directions. Due to nonlinear nature of surface water waves which is described by free-surface B.Cs., free waves interact among themselves. Free waves: their wavenumber and frequency obey the dispersion relation. Bound Waves: their wavenumber and frequency do not satisfy the dispersion relation. Bound waves result from wave-wave interactions among free waves.

2. 4.1 Weak and Strong Wave-Wave Interaction Physical phenomena resulting from strong interactions are observable soon after free waves start to interact. Those of weak interactions become substantial only after hundreds of wave periods (Su and Green 1981; Phillips 1979). Weak interactions, also known as resonance wave interactions, may occur when the frequencies and wavelengths of interacting free waves satisfy the corresponding resonance conditions. Strong Interactions occur once interacting free waves exist in the same area.

3. Weak Interactions result in energy transfer among free waves of different frequencies or wavenumbers (Phillips 1960, Hasselmann 1962). This mechanism is crucial to wave energy transfer among free waves of different wave frequencies or wavenumbers in the context of air-sea interactions (Komen et al. 1994). Strong interactions are observable immediately after the interactions (among free waves) start. They disappear except phase shifts after the interacting free waves longer overlap (Yuen & Lake 1982). Strong interactions do not result in long- lasting effects as the weak interactions have on energy transfer among free waves.

4. 4.2 Magnitude of Wave-Wave Interactions Definitions of wave steepness of an irregular wave train:

5. Definition of magnitude in terms of wave steepness : A wave-wave interaction is in general described or dictated by one or several nonlinear forcing terms (in the free-surface boundary conditions) involving the multiplication of the amplitudes of free waves,

6. Based on this definition, 1)an interaction of second order involves two free waves; 2)an interaction of third order involves three free waves in the forcing term; 3) In general, an interaction of order involves N free waves. *These N free waves are not necessary to be N different free waves. For example, they can be only one free wave in the multiplication by multiplying itself N-1 times.

10. Interactions of fourth or higher orders: They are weaker and become significant only in very steep Waves. Also involve both strong and weak interactions. Types of Resonant Interactions : 1)Type (I) Interaction (or instability), can occur at 3 rd order, Quartet wave interaction, predominantly 2-D. 2) Type (II) Interaction, can occur at 4 th order, Quintet wave interaction, predominantly 3-D

11. 4.3 Impact of Strong Interactions on Irregular Waves In a linear spectral method, nonlinear wave interactions are ignored in the decomposition of a measured wave field as well as in the calculation of resultant wave properties. In short, bound waves are treated as free waves of the same frequency. When ocean waves are not steep, the free waves are dominant in almost the entire frequency range and a linear spectral method may be a simple and fairly good approximation. When ocean waves are steep, the free waves near the spectral peak frequency still remain dominant but the bound waves may become dominant or comparable to the free waves in the frequency ranges either much lower or higher than the peak frequency.

12. 4.4 Various Perturbation Methods Conventional Perturbation Methods Mode Coupling Method (MCM), Stokes Expansion Zakharov Equation Method (ZEM) Common Features: Linear Phases for free & bound waves. Potentials are constructed based on a separation variable method. Phase Modulation Methods (PMM): Nonlinear Phases and Potential are not constructed based on a SVM.

13. Differences Between MCM & ZEM Expansion of the free-surface boundary conditions New variables in ZEM Continuous and discrete wavenumbers. The results obtained respectively using MCM and ZEM are virtually the same (Zhang & Chen 1999).